Homogeneous sets in graphs and a chromatic multisymmetric function

In this paper, we extend the chromatic symmetric function X to a chromatic k-multisymmetric function $X_k$, defined for graphs equipped with a partition of their vertex set into k parts. We demonstrate that this new function retains the basic properties and basis expansions of X, and we give a method for systematically deriving new linear relationships for X from previous ones by passing them through $X_k$.

In particular, we show how to take advantage of homogeneous sets of G (those $S\subseteq V\left(G\right)$ such that each vertex of $V\left(G\right)﹨S$ is either adjacent to all of S or is nonadjacent to all of S) to relate the chromatic symmetric function of G to those of simpler graphs. Furthermore, we show how extending this idea to homogeneous pairs $S_1\bigsqcup S_2\subseteq V\left(G\right)$ generalizes the process used by Guay-Paquet to reduce the Stanley-Stembridge conjecture to unit interval graphs.